Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 35–54

A GENERALIZED NEVANLINNA THEOREM FOR SUPERTEMPERATURES

Neil A. Watson

University of Canterbury, Department of Mathematics and Statistics Christchurch, New Zealand; N.Watson@math.canterbury.ac.nz

Abstract. A new proof of Nevanlinna’s first fundamental theorem for supertemperatures enables us to generalize the result to cover the case of the mean values of supertemperatures over level surfaces of Green functions generally. Generalizations in another direction are also obtained.

1. Introduction

In [11], an analogue for supertemperatures of Nevanlinna’s First Fundamental Theorem for superharmonic functions was presented. A new and much simpler proof has now been discovered, which enables us to generalize the theorem to include the general case of the mean values of supertemperatures over level surfaces of Green functions considered in [10]. This is the result of Theorem 1 below, and it is followed by an application to thermic majorization in Theorem 2. Subsequently, all the theorems in [11] are generalized in this way, and those which are concerned with supertemperatures on lower half-spaces R^{n}×]−∞, a[ are extended to results on sets of the form Λ(p_{0}, D) which appear in the statement of the strong minimum principle, where D is any open set that is Dirichlet regular for the adjoint heat operator ∑_{i=1}^{n} D_{i}^{2} + D_{t}.

Generalizations in another direction are also obtained. Instead of the quotients of surface mean values that appear in [11, Theorem 2], we use quotients of differences of the generalized mean values below. This renders the condition “v(p_{0}) = ∞” redundant, and the corresponding infinity conditions of [11, Theorems 3 and 4] are also absent from the generalizations below. One consequence of this is that, whereas [11, Theorem 4] gave Hausdorff measure estimates of certain polar sets, in the corresponding result below the sets need not be polar.

We work in R^{n+1} , a typical point of which we usually denote by p or q, and rarely by (x, t) with x ∈ R^{n} and t ∈ R. We denote by D an open subset of R^{n+1} that is Dirichlet regular for the adjoint heat operator, and by G_{D} its Green function for the heat operator ∑_{i=1}^{n} D_{i}^{2} − D_{t} (but in the case D = R^{n+1} we omit the subscript). All our positive measures are locally finite Borel measures, and our signed measures are differences of pairs of positive measures. The terms ‘positive’ and ‘increasing’ are used in the wide sense. Given a positive measure μ on D, its Green potential G_{D}μ is defined by

2000 Mathematics Subject Classification: Primary 31C05, 35K05.

G_{D}μ(p) = ∫_{D} G_{D}(p, q) dμ(q)

for all p ∈ D. A temperature is a solution of the heat equation, and a supertemperature is the corresponding analogue of a superharmonic function. A thermic minorant of a supertemperature w is a temperature u such that u ≤ w. We denote by E an open subset of R^{n+1} (that is not necessarily Dirichlet regular). If w is a supertemperature on E, then the Riesz Decomposition Theorem associates with w a positive measure on E, which we call the Riesz measure for w. (See [6] and [7], or [3], for details.) If v is also a supertemperature on E, and u = v − w whenever the difference is defined (hence on E less a polar set), then u is called a δ -subtemperature on E. If ν and ω are the Riesz measures for v and w respectively, then ν − ω will be called the Riesz (signed) measure for u. The Riesz measure for u is uniquely determined [11].

For all c > 0, we put τ(c) = (4πc)^{−n/2} . Let p_{0} ∈ D. There is a positive, bounded solution h of the adjoint heat equation on D such that G_{D}(p_{0}, ·) = G(p_{0}, ·) − h, so that G_{D}(p_{0}, ·) is infinitely differentiable on D\{p_{0}}. It therefore follows from Sard’s Theorem ([4, p. 45]) that, for almost every c > 0, the set {p ∈ D : G_{D}(p_{0}, p) = τ(c)} is a smooth regular n-dimensional manifold. We call such a value of c a regular value. We put

Ω_{D}(p_{0}, c) = {p ∈ D : G_{D}(p_{0}, p) > τ(c)}

and

\overline{Ω}′_{D}(p_{0}, c) = \overline{Ω}_{D}(p_{0}, c)\\{p_{0}\},

omitting the subscript if D = R^{n+1} (in which case Ω(p_{0}, c) is the heat ball with centre p_{0} and radius c). Because G_{D}(p_{0}, ·) is a solution of the adjoint equation on D\{p_{0}}, each set Ω_{D}(p_{0}, c) is Dirichlet regular for that equation. For any regular value of c,

∂Ω_{D}(p_{0}, c) = {p ∈ D : G_{D}(p_{0}, p) = τ(c)} ∪ {p_{0}}.

Since G_{D}(p_{0}, ·) is lower semicontinuous on D, each Ω_{D}(p_{0}, c) is an open set. The assumption that D is Dirichlet regular for the adjoint operator implies that G_{D}(p_{0}, ·) can be continuously extended to zero on ∂D, so that \overline{Ω}_{D}(p_{0}, c) ⊆ D for all c > 0. Furthermore, Ω_{D}(p_{0}, c) is bounded and connected ([10, p. 167]). If c is a regular value, then the outward unit normal ν = (ν_{x}, ν_{t}) to ∂Ω_{D}(p_{0}, c) is given by the standard formula ν = −∇G_{D}(p_{0}, ·)∥∇G_{D}(p_{0}, ·)∥^{−1} . Therefore, if ∇_{x}u = (D_{1}u, …, D_{n}u) denotes the gradient in the spatial variables only, we have

⟨∇_{x}G_{D}(p_{0}, ·), ν_{x}⟩ = −∥∇_{x}G_{D}(p_{0}, ·)∥^{2}∥∇G_{D}(p_{0}, ·)∥^{−1}.

This function is bounded on ∂Ω_{D}(p_{0}, c)\{p_{0}} (see [10, pp. 167–8]). The surface mean value over ∂Ω_{D}(p_{0}, c) is defined by

ℳ_{D}(u, p_{0}, c) = ∫_{∂Ω_{D}(p_{0},c)} −⟨∇_{x}G_{D}(p_{0}, ·), ν_{x}⟩u dσ

whenever the integral exists; here σ denotes surface area measure.

2. The generalized Nevanlinna theorem

In this section we present our generalization of [11, Theorem 1], along with some immediate consequences.

Theorem 1. Let E be an open set, let D be an open superset of E that is Dirichlet regular for the adjoint heat operator, let p_{0} ∈ E, and let c and d be regular values such that 0 < c ≤ d and \overline{Ω}_{D}(p_{0}, d) ⊆ E. If w is a supertemperature on E with Riesz measure μ, then

(1)  ℳ_{D}(w, p_{0}, c) = ℳ_{D}(w, p_{0}, d) − ∫_{c}^{d} τ′(γ)μ(\overline{Ω}′_{D}(p_{0}, γ)) dγ

and

(2)  w(p_{0}) = ℳ_{D}(w, p_{0}, d) − ∫_{0}^{d} τ′(γ)μ(\overline{Ω}′_{D}(p_{0}, γ)) dγ.

Proof. Let V be a bounded open set such that \overline{Ω}_{D}(p_{0}, d) ⊆ V and \overline{V} ⊆ E. By adding a constant if necessary, we may suppose that w ≥ 0 on V . Then w can be written in the form w = Gμ_{V} + u on V , where μ_{V} is the restriction of μ to V extended by zero to R^{n+1} . Since μ_{V} is finite, Gμ_{V} is a supertemperature on R^{n+1} . By [7, Theorem 19], there is a temperature v on V such that Gμ_{V} = G_{D}μ_{V} +v on D, so that w = G_{D}μ_{V} +h on V , where h = u+v. Now observe that the means in (1) are finite (by [10, Theorem 2], or [1], [2]), and that ℳ_{D}(h, p_{0}, c) = h(p_{0}) (by [10, Theorem 1]). It follows that

ℳ_{D}(w, p_{0}, c) − ℳ_{D}(w, p_{0}, d) = ℳ_{D}(G_{D}μ_{V} , p_{0}, c) − ℳ_{D}(G_{D}μ_{V} , p_{0}, d)
= ∫_{V} (ℳ_{D}(G_{D}( · , q), p_{0}, c)
− ℳ_{D}(G_{D}( · , q), p_{0}, d)) dμ(q)

= ∫_{V} ((τ(c) ∧ G_{D}(p_{0}, q)) − (τ(d) ∧ G_{D}(p_{0}, q))) dμ(q) in view of [2, Theorem 2 Corollary], and [1]. By definition of Ω_{D}(p_{0}, c), we have τ(c) ∧ G_{D}(p_{0}, q) = τ(c) if and only if q ∈ \overline{Ω}′_{D}(p_{0}, c), so that

(τ(c) ∧ G_{D}(p_{0}, q)) − (τ(d) ∧ G_{D}(p_{0}, q))

= \{ \begin{matrix}τ(c) − τ(d) & if q ∈ \overline{Ω}′_{D}(p_{0}, c), \\ G_{D}(p_{0}, q) − τ(d) & if q ∈ \overline{Ω}_{D}(p_{0}, d)\\overline{Ω}_{D}(p_{0}, c), \\ 0 & if q ∉ \overline{Ω}′_{D}(p_{0}, d). \end{matrix}

Hence

ℳ_{D}(w, p_{0}, c) − ℳ_{D}(w, p_{0}, d) = ∫_{Ω_{D}(p_{0},d)}((τ(c) ∧ G_{D}(p_{0}, q)) − τ(d)) dμ(q).

If we now put λ(γ) = μ(\overline{Ω}′_{D}(p_{0}, γ)) whenever 0 < γ ≤ d, we have

ℳ_{D}(w, p_{0}, c) − ℳ_{D}(w, p_{0}, d) = ∫_{0}^{d}((τ(c) ∧ τ(γ)) − τ(d)) dλ(γ)

= ((τ(c) ∧ τ(γ)) − τ(d))λ(γ) |_{0}^{d} − ∫_{c}^{d} τ′(γ)λ(γ) dγ

= − ∫_{c}^{d} τ′(γ)μ(\overline{Ω}′_{D}(p_{0}, γ)) dγ,

which proves (1). Making c → 0 in (1), we obtain (2).

The corollaries of [11, Theorem 1] can also be extended to the present situation.

Corollary 1. Let E be an open set, let D be an open superset of E that is Dirichlet regular for the adjoint heat operator, and let p_{0} ∈ E. If w is a supertemperature on E, then ℳ_{D}(w, p_{0}, c) = o(τ(c)) as c → 0 through regular values.

The proof is similar to that of the case D = R^{n+1} in [11].

Corollary 2. Let D be an open set which is Dirichlet regular for the adjoint heat operator, let p_{0} ∈ D, let w be a positive supertemperature on an open superset of Λ(p_{0}, D) ∪ {p_{0}}, and let μ be the Riesz measure for w. Then

τ(c)μ(\overline{Ω}′_{D}(p_{0}, c)) ≤ ℳ_{D}(w, p_{0}, c)

for all regular values of c.

Proof. Note that G_{D}(p_{0}, q) > 0 if and only if q ∈ Λ(p_{0}, D) (see [7, Theorem 14], and [3, p. 300]). Therefore \overline{Ω}_{D}(p_{0}, c) ⊆ Λ(p_{0}, D) ∪ \{p_{0}\} for all c > 0. It now follows from Theorem 1 that, if c and d are regular values with c < d, then

ℳ_{D}(w, p_{0}, c) ≥ − ∫_{c}^{d} τ′(γ)μ(\overline{Ω}′_{D}(p_{0}, γ)) dγ
≥ μ(\overline{Ω}′_{D}(p_{0}, c))(τ(c) − τ(d))
→ τ(c)μ(\overline{Ω}′_{D}(p_{0}, c))

as d → ∞.

Corollary 3. Let D be an open set which is Dirichlet regular for the adjoint heat operator, let p_{0} ∈ D, and let μ be the Riesz measure for a positive supertemperature w on E ∈ {D, Λ(p_{0}D)}. Then

lim_{c→∞} τ(c)μ(\overline{Ω}′_{D}(p, c)) = 0

for all p ∈ E.

Proof. By a result in [9], the domain Λ(p_{0}, D) is Dirichlet regular for the adjoint heat operator. Furthermore, the Green function for Λ(p_{0}, D) is the restriction of G_{D} to Λ(p_{0}, D) × Λ(p_{0}, D) (by [7, Theorem 14], and [3, p. 300]), so that Ω_{Λ(p_{0},D)}(p, c) = Ω_{D}(p, c) for all p ∈ Λ(p_{0}, D). It therefore suffices to prove the result with E = D. The greatest thermic minorant u of w is given by

u(p) = lim_{c→∞} ℳ_{D}(w, p, c)

for all p ∈ D (see [10, Theorem 7], and [8]). Since μ is also the Riesz measure for w − u, it therefore follows from Corollary 2 and [10, Theorem 1] that

τ(c)μ(\overline{Ω}′_{D}(p_{0}, c)) ≤ ℳ_{D}(w − u, p, c) = ℳ_{D}(w, p, c) − u(p) → 0

as c → ∞ through regular values. Hence, given ε > 0 we can find K such that μ(\overline{Ω}′_{D}(p_{0}, c)) ≤ ετ(c)^{−1} for all regular values of c > K, and hence for all c > K because μ(\overline{Ω}′_{D}(p_{0}, ·)) is an increasing function of c, and τ is continuous.

Theorem 1 can also be used to extend [10, Theorem 7].

Theorem 2. Let D be an open set which is Dirichlet regular for the adjoint heat operator, and let w be a supertemperature on D. Then the following statements are equivalent:

(i) w has a thermic minorant on D. (ii) There is a sequence {p_{j}} in D such that D = ⋃_{j=1}^{∞} Λ(p_{j}, D) and for each j

the function ℳ_{D}(w, p_{j}, ·) is bounded below on the set of all regular values. (iii) There is a sequence {p_{j}} in D such that D = ⋃_{j=1}^{∞} Λ(p_{j}, D) and

(3)  ∫_{1}^{∞} γ^{−(n+2)/2}μ(\overline{Ω}′_{D}(p_{j}, γ)) dγ < ∞

for all j, where μ is the Riesz measure for w.

Proof. The equivalence of (i) and (ii) is [10, Theorem 7]. To prove that (ii) and (iii) are equivalent, let {p_{j}} be a sequence in D such that D = ⋃_{j=1}^{∞} Λ(p_{j}, D). Given j, if c and d are regular values and c < d, then

ℳ_{D}(w, p_{j}, c) = ℳ_{D}(w, p_{j}, d) − τ′(1) ∫_{c}^{d} γ^{−(n+2)/2}μ(\overline{Ω}′_{D}(p_{j}, γ)) dγ

by Theorem 1. Furthermore, the means are finite valued ([10, Theorem 2], or [2]). Therefore, if we fix c and make d → ∞, we see that ℳ_{D}(w, p_{j}, ·) is bounded below if and only if (3) holds.

3. The behaviour of the means for small regular values

Theorem 3 below generalizes [11, Theorem 2] in two directions. First it removes the restriction v(p_{0}) = ∞, and second it replaces ℳ by ℳ_{D}.

We need some notation. Let E be an open set, and let D be an open superset of E that is Dirichlet regular for the adjoint heat operator. If \overline{Ω}_{D}(p_{0}, d) ⊆ E, ν is a positive measure on E, and 0 ≤ b < c ≤ d, we put

I_{ν,D}(p_{0}; b, c) = − ∫_{b}^{c} τ′(γ)ν(\overline{Ω}′_{D}(p_{0}, γ)) dγ = κ_{n} ∫_{b}^{c} γ^{−(n+2)/2}ν(\overline{Ω}′_{D}(p_{0}, γ)) dγ,

where κ_{n} = −τ′(1) = n2^{−n−1}π^{−n/2} .

We include for completeness the definition of

lim sup_{0<b<c→0} f(b, c),

although it is the natural one. Those of the corresponding lim inf and lim are then obvious.

Definition. Suppose that f(b, c) is defined as an extended-real number for Lebesgue almost all b and c such that 0 < b < c < d, and that l ∈ R. We write

lim sup_{0<b<c→0} f(b, c) = l

if to each ε > 0 there corresponds δ > 0 such that f(b, c) < l + ε whenever f(b, c) is defined with 0 < b < c < δ, and there is a sequence {(b_{k}, c_{k})} such that 0 < b_{k} < c_{k} → 0 and f(b_{k}, c_{k}) → l as k → ∞. We also write

lim sup_{0<b<c→0} f(b, c) = ∞

if there is a sequence {(b_{k}, c_{k})} such that 0 < b_{k} < c_{k} → 0 and f(b_{k}, c_{k}) → ∞. Finally, we write

lim sup_{0<b<c→0} f(b, c) = −∞

if to each A ∈ R there corresponds δ > 0 such that f(b, c) < A whenever f(b, c) is defined with 0 < b < c < δ.

Theorem 3. Let E be an open set, let D be an open superset of E that is Dirichlet regular for the adjoint heat operator, let u be a δ -subtemperature on E with Riesz measure μ, and let ν be a positive measure on E. Then

(4)  lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{I_{ν,D}(p; b, c)} ≤ lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{ν(\overline{Ω}′_{D}(p, d))}

whenever the latter exists. Furthermore, if I_{ν,D}(p; 0, c) < ∞ for all sufficiently small values of c, and u(p) is defined and finite, then

(5)  lim sup_{c→0} \frac{u(p) − ℳ_{D}(u, p, c)}{I_{ν,D}(p; 0, c)} ≤ lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{ν(\overline{Ω}′_{D}(p, d))}.

Proof. Suppose that the upper limit on the right-hand side of (4) exists, and denote it by l. If l = ∞ there is nothing to prove. Otherwise, given a real number A > l, we can find δ > 0 such that

(6)  \frac{μ(\overline{Ω}′_{D}(p, d))}{ν(\overline{Ω}′_{D}(p, d))} < A  whenever 0 < d < δ.

If ν(\overline{Ω}′_{D}(p, d)) = 0 for all d < η(≤ δ), then (6) can hold only if μ(\overline{Ω}′_{D}(p, d)) < 0 for all d < η. Then I_{ν,D}(p; b, c) = 0 whenever c < η, and (1) shows that

ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c) < 0

for all regular values such that 0 < b < c < η, so that (4) holds with both sides −∞. On the other hand, if ν(\overline{Ω}′_{D}(p, d)) > 0 for all d, then by (1)

\begin{aligned}\frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{I_{ν,D}(p; b, c)} & = \frac{−1}{I_{ν,D}(p; b, c)} ∫_{b}^{c} τ′(γ)ν(\overline{Ω}′_{D}(p, γ))\frac{μ(\overline{Ω}′_{D}(p, γ))}{ν(\overline{Ω}′_{D}(p, γ))} dγ \\ & < A \end{aligned}

for all regular values such that 0 < b < c < δ, and again (4) holds.

The inequality (5) can be proved in a similar way, using (2) instead of (1).

Corollary. If u is a δ -subtemperature with Riesz measure μ on an open set E, then

(7)  lim_{0<b<c→0} \frac{ℳ (u, p, b) − ℳ (u, p, c)}{c − b} = κ_{n} lim_{d→0} \frac{μ(\overline{Ω}′(p, d))}{d^{(n+2)/2}}

whenever the latter exists. Furthermore, if u(p) is defined and finite, then

lim_{c→0} \frac{ℳ (u, p, c) − u(p)}{c} = −κ_{n} lim_{d→0} \frac{μ(\overline{Ω}′(p, d))}{d^{(n+2)/2}}

whenever the latter exists.

Proof. We take D = R^{n+1} and ν to be (n+1)-dimensional Lebesgue measure in Theorem 3, noting that the reverse inequalities hold for lower limits. A routine calculation shows that ν(\overline{Ω}′(p, γ)) = λ_{n} γ^{(n+2)/2} , where

λ_{n} = 2^{n+1}(nπ)^{n/2}(n + 2)^{−(n+2)/2}.

Therefore, if 0 ≤ b < c and \overline{Ω}(p, c) ⊆ E, then

I_{ν}(p; b, c) = κ_{n}λ_{n} ∫_{b}^{c} dγ = (\frac{n}{n + 2})^{(n+2)/2}(c − b).

It now follows from (4) and (5), together with their duals for lower limits, that

lim_{0<b<c→0} \frac{ℳ (u, p, b) − ℳ (u, p, c)}{κ_{n}λ_{n}(c − b)} = lim_{d→0} \frac{μ(\overline{Ω}′(p, d))}{λ_{n}d^{(n+2)/2}}

and

lim_{c→0} \frac{u(p) − ℳ (u, p, c)}{κ_{n}λ_{n}c} = lim_{d→0} \frac{μ(\overline{Ω}′(p, d))}{λ_{n}d^{(n+2)/2}}

whenever the last limit exists.

It is not immediately apparent that Theorem 3 is a generalization of [11, Theorem 2]. To demonstrate that it is, we first write it in a different form, then deduce the earlier result as the case D = R^{n+1} of the subsequent corollary.

Theorem 4. Let E be an open set, and let D be an open superset of E that is Dirichlet regular for the adjoint operator. Let u be a δ -subtemperature with Riesz measure μ, and v a supertemperature with Riesz measure ν, on E. Then

(8)  lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{ℳ_{D}(v, p, b) − ℳ_{D}(v, p, c)} ≤ lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{ν(\overline{Ω}′_{D}(p, d))}

whenever the latter exists. Furthermore, if u(p) is defined and finite, and v(p) < ∞, then

lim sup_{c→0} \frac{u(p) − ℳ_{D}(u, p, c)}{v(p) − ℳ_{D}(v, p, c)} ≤ lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{ν(\overline{Ω}′_{D}(p, d))}.

Proof. In view of Theorem 1 and the finiteness of the mean values, the result follows from Theorem 3.

Corollary. If the hypotheses of Theorem 4 are satisfied and v(p) = ∞, then

(9)  lim sup_{c→0} \frac{ℳ_{D}(u, p, c)}{ℳ_{D}(v, p, c)} ≤ lim sup_{c→0} \frac{μ(\overline{Ω}′_{D}(p, c))}{ν(\overline{Ω}′_{D}(p, c))}.

Proof. Let l denote the left-hand side of (8). We prove that

(10)  lim sup_{c→0} \frac{ℳ_{D}(u, p, c)}{ℳ_{D}(v, p, c)} ≤ l,

which implies that (9) holds, in view of (8). We may assume that l < ∞. Given a real number A > l, we choose δ > 0 such that

\frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{ℳ_{D}(v, p, b) − ℳ_{D}(v, p, c)} < A

whenever b and c are regular values such that 0 < b < c < δ. By [10, Theorem 2], ℳ_{D}(v, p, d) → v(p) as d → 0 (through regular values), so that our hypothesis v(p) = ∞ means we may suppose that ℳ_{D}(v, p, d) > 0 for all regular values of d < δ. With this assumption, we fix a regular value of c < δ. Given ε > 0, we choose η < c such that both

\frac{ℳ_{D}(v, p, c)}{ℳ_{D}(v, p, b)} < ε  and  \frac{ℳ_{D}(u, p, c)}{ℳ_{D}(v, p, b)} < ε

for all regular values of b < η. Then

\begin{aligned}\frac{ℳ_{D}(u, p, b)}{ℳ_{D}(v, p, b)} & = \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{ℳ_{D}(v, p, b) − ℳ_{D}(v, p, c)} (1 − \frac{ℳ_{D}(v, p, c)}{ℳ_{D}(v, p, b)} ) + \frac{ℳ_{D}(u, p, c)}{ℳ_{D}(v, p, b)} \\ & < max\{A, (1 − ε)A\} + ε \end{aligned}

if b < η. This proves (10), and (9) follows.

We can also generalize [11, Theorem 3] in a similar way.

Theorem 5. Let E be an open set, let D be an open superset of E that is Dirichlet regular for the adjoint operator, and let u be a δ -subtemperature with Riesz measure μ on E. Let α > 0, let f be a positive, increasing, absolutely continuous function on [0, α], and let

f̂(b, c) = κ_{n} ∫_{b}^{c} γ^{−(n+2)/2}f(γ) dγ

whenever 0 ≤ b < c ≤ α. Then

(11)  lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{f̂(b, c)} ≤ lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{f(d)}

for every p ∈ E. Furthermore, if u(p) is defined and finite, and f̂(0, c) < ∞ for all sufficiently small values of c, then

(12)  lim sup_{c→0} \frac{u(p) − ℳ_{D}(u, p, c)}{f̂(0, c)} ≤ lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{f(d)}.

Proof. If f(0) ≠ 0, then μ(\overline{Ω}′_{D}(p, d)) = o(f(d)) as d → 0, so we have to prove that

lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{f̂(b, c)} = 0.

(Note that in this case f̂(0, c) = ∞, so that the conditions for (12) are not satisfied.) If 0 < b < c < α, then

f̂(b, c) ≥ f(0) ∫_{b}^{c} −τ′(γ) dγ = f(0)(τ(b) − τ(c)),

so that

0 ≤ \frac{τ(b) − τ(c)}{f̂(b, c)} ≤ \frac{1}{f(0)}.

Also, by Theorem 1,

\frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{τ(b) − τ(c)} = \frac{1}{τ(b) − τ(c)} ∫_{b}^{c} −τ′(γ)μ(\overline{Ω}′_{D}(p, γ)) dγ,

which is o(1) as 0 < b < c → 0 because μ(\overline{Ω}′_{D}(p, γ)) = o(1) as γ → 0. It follows that

\frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{f̂(b, c)} = (\frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{τ(b) − τ(c)} )(\frac{τ(b) − τ(c)}{f̂(b, c)}) = o(1)

as 0 < b < c → 0 through regular values.

Now consider the case where f(0) = 0. We choose d ≤ α such that \overline{Ω}_{D}(p, d) is contained in E, and define a measure ν on E by putting

dν = −∥∇_{x}G_{D}(p, ·)∥^{2}(\frac{f′}{τ′} )(\frac{G_{D}(p, ·)^{−2/n}}{4π} )χ_{Ω_{D}(p,d)} dλ,

where χ_{A} denotes the characteristic function of a set A, and λ denotes (n + 1)- dimensional Lebesgue measure. If 0 < c < d, it follows from results in [10, pp. 167–70] that

ν(\overline{Ω}′_{D}(p, c)) = − ∫_{Ω_{D}(p,c)} ∥∇_{x}G_{D}(p, ·)∥^{2}(\frac{f′}{τ′} )(\frac{G_{D}(p, ·)^{−2/n}}{4π} ) dλ

= ∫_{0}^{c}(∫_{∂Ω_{D}(p,γ)} \frac{∥∇_{x}G_{D}(p, ·)∥^{2}}{∥∇G_{D}(p, ·)∥} f′(γ) dσ) dγ

= ∫_{0}^{c} ℳ_{D}(1, p, γ)f′(γ) dγ

= ∫_{0}^{c} f′(γ) dγ = f(c),

so that

I_{ν,D}(p; b, c) = − ∫_{b}^{c} τ′(γ)f(γ) dγ = f̂(b, c)

whenever 0 ≤ b < c ≤ d. The inequalities (11) and (12) now follow from Theorem 3.

The special case D = R^{n+1} of the following corollary is [11, Theorem 3].

Corollary 1. If the hypotheses of Theorem 5 are satisfied and f̂(0, α) = ∞, then

lim sup_{b→0} \frac{ℳ_{D}(u, p, b)}{f̂(b, α)} ≤ lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{f(d)}

for each p ∈ E.

Proof. Given p ∈ E, let l denote the left-hand side of (11). In view of (11), it is more than enough to prove that

(13)  lim sup_{b→0} \frac{ℳ_{D}(u, p, b)}{f̂(b, α)} ≤ l.

We may assume that l < ∞. Given any real number A > l, we can find δ > 0 such that

\frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{f̂(b, c)} < A

whenever b and c are regular values such that 0 < b < c < δ. Fix c < δ. Given ε > 0, we choose η < c such that both

\frac{ℳ_{D}(u, p, c)}{f̂(b, α)} < ε  and  \frac{f̂(c, α)}{f̂(b, α)} < ε

whenever 0 < b < η. Then

\begin{aligned}\frac{ℳ_{D}(u, p, b)}{f̂(b, α)} & = \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{f̂(b, c)} (1 − \frac{f̂(c, α)}{f̂(b, α)}) + \frac{ℳ_{D}(u, p, c)}{f̂(b, α)} \\ & < max\{A, (1 − ε)A\} + ε \end{aligned}

for every regular value of b < η. The inequality (13) follows.

The extra generality of Theorem 5 over its first corollary enables us to generalize [11, Theorem 3 Corollary], and remove its restriction on the range of values of its parameter β, as follows.

Corollary 2. Let E be an open set, let D be an open superset of E that is Dirichlet regular for the adjoint operator, let u be a δ -subtemperature with Riesz measure μ on E, and let p ∈ E. Then

(\frac{n − β}{2} ) lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{b^{−(n−β)/2} − c^{−(n−β)/2}} ≤ κ_{n} lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{d^{β/2}}

if 0 ≤ β < n,

lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{log(c/b)} ≤ κ_{n} lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{d^{n/2}},

and

(\frac{β − n}{2}) lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{c^{(β−n)/2} − b^{(β−n)/2}} ≤ κ_{n} lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{d^{β/2}}

if β > n.

Proof. We take f(d) = d^{β/2} for β ≥ 0, in Theorem 5. Then

f̂(b, c) = κ_{n} ∫_{b}^{c} γ^{(β−n−2)/2} dγ,

which is equal to κ_{n} times

2(b^{−(n−β)/2} − c^{−(n−β)/2})/(n − β) if 0 ≤ β < n,
log(c/b)  if β = n,
2(c^{(β−n)/2} − b^{(β−n)/2})/(β − n)  if β > n,

and the result follows.

4. The parabolic Hausdorff measures of certain sets

We use Theorem 5 to study the size of the sets of points p where

\frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{f̂(b, c)}

is unbounded as 0 < b < c → 0, for a given function f and supertemperature u. The size is estimated in terms of the parabolic Hausdorff measures discussed in [5], which have the appropriate mixed homogeneity.

We recall the necessary definitions. Let h be an increasing function on ]0, ∞[ such that h(0+) = 0. Let 𝒫 denote the class of all sets of the form

(∏_{i=1}^{n}[a_{i}, a_{i} + r]) × [a, a + r^{2}].

The set of this form which is centred at p is denoted by P (p, r). For an arbitrary set S, the outer parabolic h-measure of S is defined by

𝒫 − h − m_{∗}(S) = lim_{δ→0+} inf{∑_{i=1}^{∞} h(diam P_{i}) : P_{i} ∈ 𝒫, E ⊆ ⋃_{i=1}^{∞} P_{i}, diam P_{i} < δ}.

The associated measure, defined on a σ -field that contains the Borel sets, is denoted by 𝒫 − h − m. When h(s) = s^{α} for some α > 0, we write 𝒫 − Λ^{α} − m for 𝒫 − h − m.

Theorem 6. Let E be an open set, let D be an open superset of E that is Dirichlet regular for the adjoint operator, and let u be a supertemperature on E. Let 0 < α < ∞, and let h be an increasing function on [0, ∞[ that is absolutely continuous on [0, \sqrt{α} ] and satisfies h(0) = 0, h(2s) ≤ Kh(s) for all s > 0, where K is a constant. Put

F (b, c) = κ_{n} ∫_{b}^{c} γ^{−(n+2)/2}h(\sqrt{γ} ) dγ

whenever 0 < b < c ≤ α.

(i) The set

(14)  \{p ∈ E : lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{F (b, c)} = ∞\}

has 𝒫 − h-measure zero. (ii) If, in addition,

∫_{0}^{√\overline{α}} s^{−n−1}h(s) ds = ∞,

then the set

(15)  \{p ∈ E : lim sup_{b→0} \frac{ℳ_{D}(u, p, b)}{F (b, α)} = ∞\}

also has 𝒫 − h-measure zero.

Proof. The sets (14) and (15) are subsets of

(16)  \{p ∈ E : lim sup_{d→0} \frac{μ(\overline{Ω}′_{D}(p, d))}{h(\sqrt{d} )} = ∞\},

by Theorem 5 and its Corollary 1 (with f(γ) = h(\sqrt{γ} )). Furthermore, since G_{D} ≤ G on D (by [7, Theorem 10]), we have Ω_{D}(p, d) ⊆ Ω(p, d). Therefore, given d > 0, if we choose r = 3\sqrt{nd/e} then \overline{Ω}′_{D}(p, d) ⊆ P (p, r). It follows that the set (16) is a subset of

(17)  \{p ∈ E : lim sup_{r→0} \frac{μ(P (p, r))}{h(δr)} = ∞\},

where δ = \sqrt{e/9n} < 1. If i is chosen so that 2^{i}δ > 1, then

h(δr) ≥ K^{−i}h(2^{i}δr) ≥ K^{−i}h(r).

Therefore the set (17) is a subset of

\{p ∈ E : lim sup_{r→0} \frac{μ(P (p, r))}{h(r)} = ∞\},

which has 𝒫 − h-measure zero by the lemma in [11].

Remark. The case D = R^{n+1} of Theorem 6(ii) was proved in [11].

Corollary. Let E be an open set, let D be an open superset of E that is Dirichlet regular for the adjoint operator, and let u be a supertemperature on E.

(i) If 0 < β < n, then the set

\{p ∈ E : lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{b^{−(n−β)/2} − c^{−(n−β)/2}} = ∞\}

has 𝒫 − Λ^{β} -measure zero. (ii) The set

\{p ∈ E : lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{log(c/b)} = ∞\}

has 𝒫 − Λ^{n} -measure zero. (iii) If n < β ≤ n + 2, then the set

(18)  \{p ∈ E : lim sup_{0<b<c→0} \frac{ℳ_{D}(u, p, b) − ℳ_{D}(u, p, c)}{c^{(β−n)/2} − b^{(β−n)/2}} = ∞\}

has 𝒫 − Λ^{β} -measure zero.

Proof. The result is obtained by taking h(s) = s^{β} for β ∈]0, n + 2] in Theorem 6. (Note that 𝒫−Λ^{β} −m(S) = 0 for all subsets S of R^{n+1} if β > n+2.)

Remark. In Theorem 6(ii), the extra condition on h ensures that F (b, α) → ∞ as b → 0, so that u(p) = ∞ for all p in the set (15); thus the set is polar. In Theorem 6(i), polarity is not so readily determined, and whether or not the set (14) is polar depends upon h. By [5, Theorem 1], if a set is not polar then its 𝒫 − Λ^{n} -measure must be strictly positive, and so the sets considered in parts (i) and (ii) of the corollary are polar. The set in part (iii), however, may not be polar, as the following example shows.

Example. Given β such that n < β ≤ n + 2, put α = \frac{1}{3}(n + 3 − β), so that \frac{1}{3} ≤ α < 1. Let t_{0} ∈ R, Q = ∏_{i=1}^{n}]0, 1[, S = Q×]t_{0} − 1, t_{0}[, and

μ(B) = ∫∫_{B∩S}(t_{0} − t)^{−α} dx dt

for every Borel subset B of R^{n+1} . Then μ is a finite measure, so that Gμ is a positive supertemperature on R^{n+1} . We consider the case D = E = R^{n+1} of the above corollary. Given any x_{0} ∈ Q, we put p_{0} = (x_{0}, t_{0}) and can find c_{0} > 0 such that Ω(p_{0}, c_{0}) ⊆ S. Then, whenever c < c_{0} , we have

μ(\overline{Ω}′(p_{0}, c)) = ∫∫_{Ω(p_{0},c)}(t_{0} − t)^{−α} dx dt = C_{n,α}c^{(n+2−2α)/2},

where C_{n,α} = 2^{n+1}(πn)^{n/2}(n + 2 − α)^{−(n+2)/2} . Therefore

c^{−β/2}μ(\overline{Ω}′(p_{0}, c)) = C_{n,α}c^{(n+2−β−2α)/2} → ∞ as c → 0,

because

\frac{n + 2 − β − 2α}{2} = \frac{n − β}{6} < 0.

Therefore, by Theorem 5, Corollary 2,

\frac{ℳ (u, p_{0}, b) − ℳ (u, p_{0}, c)}{c^{(β−n)/2} − b^{(β−n)/2}} → ∞

as 0 < b < c → 0, so that the set (18) contains Q × {t_{0}}, which is not polar.

5. The Riesz measures of supertemperatures on Dirichlet regular sets

In this section we generalize [11, Theorem 5] from the case of a lower halfspace R^{n}×] − ∞, a[ to that of an arbitrary open set which is Dirichlet regular for the adjoint operator.

Theorem 7. Let D be an open set which is Dirichlet regular for the adjoint heat operator, let p_{0} ∈ D, let E ∈ {D, Λ(p_{0}, D)}, and let μ be a positive measure on E.

(i) If μ is the Riesz measure of a supertemperature which has a thermic minorant on E, then

(19)  ∫_{1}^{∞} γ^{−(n+2)/2}μ(\overline{Ω}′_{D}(p, γ)) dγ < ∞

for all p ∈ E. (ii) Conversely, if there is p ∈ E such that (19) holds, then G_{D}μ is a supertem-

perature on Λ(p, D). If, in addition,

∫_{0}^{1} γ^{−(n+2)/2}μ(\overline{Ω}′_{D}(p, γ)) dγ < ∞,

then G_{D}μ(p) < ∞.

Proof. By a result in [9], the domain Λ(p_{0}, D) is Dirichlet regular for the adjoint heat operator. Furthermore, the Green function for Λ(p_{0}, D) is the restriction of G_{D} to Λ(p_{0}, D) × Λ(p_{0}, D) (by [7, Theorem 14], or [3, p. 300]), and Λ(p, Λ(p_{0}, D)) = Λ(p, D) for any p ∈ Λ(p_{0}, D). It therefore suffices to prove the result when E = D.

(i) Let w be a supertemperature which has a thermic minorant u on D, and whose Riesz measure is μ. Then μ is also the Riesz measure for w −u. Therefore, if p ∈ D and c, d are regular values such that c < d, Theorem 1 shows that

\begin{aligned}ℳ_{D}(w − u, p, c) & = ℳ_{D}(w − u, p, d) + κ_{n} ∫_{c}^{d} γ^{−(n+2)/2}μ(\overline{Ω}′_{D}(p, γ)) dγ \\ & ≥ κ_{n} ∫_{c}^{d} γ^{−(n+2)/2}μ(\overline{Ω}′_{D}(p, γ)) dγ. \end{aligned}

Making d → ∞ we obtain (19), because ℳ_{D}(w − u, p, c) < ∞.

(ii) Suppose that (19) holds for some p = p_{1} ∈ D. Let {k_{j}} be an unbounded increasing sequence of regular values, and put

\begin{aligned}A_{D}(p_{1}; k_{1}, ∞) & = Λ(p_{1}, D)\\overline{Ω}′_{D}(p_{1}, k_{1}), \\ A_{D}(p_{1}; k_{1}, k_{j}) & = Ω_{D}(p_{1}, k_{j})\\overline{Ω}′_{D}(p_{1}, k_{1}) for j > 1. \end{aligned}

If u = G_{D}μ, then for all p ∈ D we put

\begin{aligned}u(p) & = ∫_{\overline{Ω}′_{D}(p_{1},k_{1})} G_{D}(p, q) dμ(q) + ∫_{A_{D}(p_{1};k_{1},∞)} G_{D}(p, q) dμ(q) \\ & = v_{1}(p) + v_{2}(p), \end{aligned}

say, and

u_{j}(p) = ∫_{A_{D}(p_{1};k_{1},k_{j})} G_{D}(p, q) dμ(q)

for j ≥ 1. Since μ is locally finite, v_{1} and every u_{j} is a supertemperature on D. Since {u_{j}} is increasing to the limit v_{2} , v_{2} is a supertemperature on Λ(p_{1}, D) if v_{2}(p_{1}) < ∞, by [6, Theorem 6]. Writing λ(γ) = μ(\overline{Ω}′_{D}(p_{1}, γ)) for all γ > 0, we have

u_{j}(p_{1}) = ∫_{k_{1}}^{k_{j}} τ(γ) dλ(γ) = [τ(γ)λ(γ)]_{k_{1}}^{k_{j}} − ∫_{k_{1}}^{k_{j}} τ′(γ)λ(γ) dγ.

Since (19) holds when p = p_{1} , we have

λ(c)τ(c) = λ(c) ∫_{c}^{∞} −τ′(γ) dγ ≤ − ∫_{c}^{∞} τ′(γ)λ(γ) dγ → 0

as c → ∞. Therefore

v_{2}(p_{1}) = lim_{j→∞} u_{j}(p_{1}) = −τ(k_{1})λ(k_{1}) − ∫_{k_{1}}^{∞} τ′(γ)λ(γ) dγ < ∞,

so that v_{2} , and hence u, is a supertemperature on Λ(p_{1}, D).

For the last part, let {c_{j}} be a decreasing null sequence of regular values (relative to p_{1} ). Then

u(p_{1}) = lim_{j→∞} ∫_{c_{j}}^{∞} τ(γ) dλ(γ)

= lim_{j→∞}(−τ(c_{j})λ(c_{j}) − ∫_{c_{j}}^{∞} τ′(γ)λ(γ) dγ)

≤ − ∫_{0}^{∞} τ′(γ)λ(γ) dγ < ∞.

6. Differences of positive supertemperatures on Λ(p_{0}, D)

Let u be a δ -subtemperature on an open set E, and let D be an open superset of E that is Dirichlet regular for the adjoint operator. If μ is the Riesz measure for u, then μ can be written minimally as a difference μ^{+} − μ^{−} of two positive measures on E. Whenever \overline{Ω}_{D}(p, c) ⊆ E, we put

λD^{+}(p, c) = μ^{+}(\overline{Ω}′_{D}(p, c)), ND^{+}(p, c) = − ∫_{0}^{c} τ′(γ)λD^{+}(p, γ) dγ,

and similarly for μ^{−}. We say that u(p_{0}) is finite if ND^{+}(p_{0}, ·) and ND^{−}(p_{0}, ·) are both finite-valued, in which case it follows from (2) that u is the difference of two supertemperatures which are finite at p_{0} . If u(p_{0}) is finite, we define the characteristic T_{D} of u at p_{0} by

T_{D}(u, p_{0}, c) = ℳ_{D}(u^{+}, p_{0}, c) + ND^{+}(p_{0}, c) − u(p_{0})

for each regular value of c such that \overline{Ω}_{D}(p_{0}, c) ⊆ E. We can use T_{D} to characterize those δ -subtemperatures on Λ(p_{0}, D) or D that can be written as a difference of two positive supertemperatures, and thus generalize [11, Theorem 6].

Theorem 8. Let D be an open set which is Dirichlet regular for the adjoint heat operator, let p_{0} ∈ D, let E ∈ {D, Λ(p_{0}, D)}, and let u be a δ - subtemperature on E.

(i) If u = u_{1} − u_{2} is the difference of two positive supertemperatures on E, and u(p_{1}) is finite, then T_{D}(u, p_{1}, ·) is an increasing function such that 0 ≤ T_{D}(u, p_{1}, c) ≤ u_{2}(p_{1}) for all regular values of c, and there is a convex function ϕ such that T_{D}(u, p_{1}, ·) = ϕ ∘ τ.

(ii) Conversely, if there is a sequence {p_{j}} in E such that E = ⋃_{j=1}^{∞} Λ(p_{j}, D), u(p_{j}) is finite for all j, and T_{D}(u, p_{j}, ·) is bounded above on the set of all regular values for each j, then u is the difference of two positive supertemperatures on E.

Proof. (i) For i ∈ {1, 2}, let μ_{i} be the Riesz measure for u_{i}, and put

λ_{D}^{i}(p_{1}, c) = μ_{i}(\overline{Ω}′_{D}(p_{1}, c)), N_{D}^{i}(p_{1}, c) = − ∫_{0}^{c} τ′(γ)λ_{D}^{i}(p_{1}, γ) dγ

for all c > 0. Since u_{1} ≥ 0, it follows from (2) that

0 = ℳ_{D}(u1^{−}, p_{1}, c) = ℳ_{D}(u_{1}, p_{1}, c) + N_{D}^{1}(p_{1}, c) − u_{1}(p_{1}).

Since μ_{1} and μ_{2} are positive and μ = μ_{1} − μ_{2} , we have μ^{+} ≤ μ_{1} and μ^{−} ≤ μ_{2} , so that

ND^{+}(p_{1}, c) ≤ N_{D}^{1}(p_{1}, c) = u_{1}(p_{1}) − ℳ_{D}(u_{1}, p_{1}, c).

A generalized Nevanlinna theorem for supertemperatures 53

Furthermore u_{1} ≥ u^{+} , so that ℳ_{D}(u^{+}, p_{1}, c) ≤ ℳ_{D}(u_{1}, p_{1}, c). Hence

T_{D}(u, p_{1}, c) ≤ ℳ_{D}(u_{1}, p_{1}, c) + (u_{1}(p_{1}) − ℳ_{D}(u_{1}, p_{1}, c)) − u(p_{1}) = u_{2}(p_{1}).

Now let v_{2} = G_{D}μ^{−} and v_{1} = u + v_{2} . Applying (2) to each v_{i} and subtracting, we obtain

u(p_{1}) = ℳ_{D}(u, p_{1}, c) + ND^{+}(p_{1}, c) − ND^{−}(p_{1}, c),

so that

T_{D}(u, p_{1}, c) = ℳ_{D}(u^{+}, p_{1}, c) + ND^{−}(p_{1}, c) − ℳ_{D}(u, p_{1}, c)

= ℳ_{D}(u^{−}, p_{1}, c) + ND^{−}(p_{1}, c)

= ℳ_{D}(u^{−}, p_{1}, c) + v_{2}(p_{1}) − ℳ_{D}(v_{2}, p_{1}, c) = v_{2}(p_{1}) − ℳ_{D}(v_{2} − u^{−}, p_{1}, c).

Let p ∈ E. If u(p) ≥ 0, then v_{1}(p) ≥ v_{2}(p) and v_{2}(p) − u^{−}(p) = v_{2}(p) = (v_{1}∧v_{2})(p). On the other hand, if u(p) < 0 then v_{1}(p) < v_{2}(p) and v_{2}(p)−u^{−}(p) = v_{1}(p) = (v_{1} ∧ v_{2})(p). Hence

T_{D}(u, p_{1}, c) = v_{2}(p_{1}) − ℳ_{D}(v_{1} ∧ v_{2}, p_{1}, c).

Since v_{1} ∧ v_{2} is a supertemperature on E, the characteristic T_{D}(u, p_{1}, ·) is increasing and real-valued on the set of regular values (by [10, Theorem 2]), there is a convex function ϕ such that T_{D}(u, p_{1}, ·) = ϕ ∘ τ (by [10, Theorem 3]) and T_{D}(u, p_{1}, 0+) = v_{2}(p_{1}) − (v_{1} ∧ v_{2})(p_{1}) ≥ 0 (by [10, Theorem 2]).

(ii) The proof is similar to that of [11, Theorem 6(ii)].

References

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[5] Taylor, S.J., and N.A. Watson: A Hausdorff measure classification of polar sets for the heat equation. - Math. Proc. Cambridge Philos. Soc. 97, 1985, 325–344.

[6] Watson, N.A.: A theory of subtemperatures in several variables. - Proc. London Math. Soc. 26, 1973, 385–417.

[7] Watson, N.A.: Green functions, potentials, and the Dirichlet problem for the heat equation. - Proc. London Math. Soc. 33, 1976, 251–298.

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54 Neil A. Watson

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Received 22 November 2000